Formulas
I have inserted a column labeled terms. 1 represents the first arrangement of tiles, 2 represents the second arrangement of tiles, and so on. n represents the nth term. The common difference is the 2nd term – 1st term. I have titled each table of results with the symbol used to represent the shape of the spacer it relates to.
L Shape Spacer Results
T Shape Spacer Results
From the table above the results in the difference column shows a number sequence increasing by the same amount from one term to the next. Therefore, this tells me that these numbers are appearing in linear sequence. As the term sequence 4,8,12… increase by 4 from one term to the next, here I can say that the sequence has a common difference of 4.
+ Shape Spacer Results
The ‘term x common difference’ column was removed from the table above due to the fact there is no common difference for this sequence. I have worked out the rule which can be applied to get from one term to the next. This same rule can be applied to find further terms in the sequence.
I am now going to collect another set of results for further investigation. I will start with putting up tiles in a straight line, then look at two rows of tiles, then three rows of tiles and so fourth. As placing the results into table form proved successful, I will repeat the process again. Below I have designed a table, this table shows the arrangement of tiles, the number of each type of spacer used, and the total number of spacers used.
Conclusion table for square tiles (adding one at a time)
Formulas found for each type of spacer
Predictions and Testing of Formulas
My results tables on page 3, display results of tile arrangements made from using square numbers. From these results I can predict how many spacers of each type that would be needed for an arrangement of tiles measuring a 6 x 6. I can do this because the lists of numbers (my results) have produced a sequence.
Once you have recognized a pattern it easy to assume what the next consecutive term will be. I now want to be able to calculate how many spacers of each type would be needed for a tile arrangement of a much larger scale. Assuming that my formulas are correct, I can therefore apply it to any number of tiles, provided it is a square number.
Take for example a tile arrangement of 25 x 25; I want to find out how many spacers will be needed. A table below shows how many spacers will be needed.
Reading from my table, a tile arrangement of 25 x 25 I will need 4 L shape spacers, 96 T shape spacers and 576 + shape spacers. 676 tile spacers are needed in total.
Say I want to tile a room that is not exactly square; I will not be able to use these formulas. But I can refer back to page 4. Page 4 illustrates how many spacers of each kind are needed starting with a straight line of tiles, then by looking at two rows of tiles then three, etc. I have displayed only 3 different tile arrangements that are rectangular form. The first is a 1 x 2. The second is a 2 x 3 and the third is a 3 x 4. I drew these diagrams for effect, but to save time I could have referred to this table to calculate the spacers needed for these three different tile arrangements.
A table below shows my predictions for tile arrangements measuring;
- 6 x 1
- 6 x 2
- 6 x 3
- 6 x 4
- 6 x 5
- 6 x 6
Over Arching Solution
Finding a formula for any tile arrangement made up from square numbers proved successful. Although when I tried to apply the formulas from my extended rectangular investigation, to a further larger scale I was inevitably rendered unsuccessful. I soon became aware of the problem; it then became clear to me that a different method would be more appropriate for investigating how many different spacers are needed for tiling an area of rectangular form.
A typical rectangular structure measuring 4 x 3 is going to require n number of L, T, & + shape spacers. I know that the amount of L shape spacers will be 4. The amount of T & + still need further investigation. My 4 x 3 has 2 sides measuring 4 tiles and 2 sides measuring 3 tiles. I am going to use the symbol H for height and W for width;
T = 4 + 3 + 4 + 3
T = (W – 1) + (H – 1) + (W – 1) + (H – 1)
T = 2(W – 1) + 2 (H – 1)
T = 10
So for a 4 x 3: 10 T shape spacers will be needed. For the + shape spacers I am also going to use the symbols H and W to represent height and width. Just like the square; the rectangular tile arrangements are also forming a rectangle within the main rectangle arrangement of each tile design;
+ = 4 x 3
+ = (W – 1) x (H – 1)
+ = 6
Now I am going to apply my new formulas to a larger rectangular arrangement measuring 12 x 9. I have displayed my results in a small table below and over the page I have drew out my 12 x 9 tile design.
I am now delighted to say that my formulas for my rectangular investigation have proved successful, and my tile design overleaf will provide evidence to confirm this.
Conclusion
Spacers can be used if the tile is nearly perfect in square and size when compared to each other. I can illustrate this thought by saying that if the room is large there will be many tiles involved. Spacers will work very well if the tiles are uniform and the room is relatively small.
After the completion of my rectangular tile investigation, it soon became apparent to me that the formulas for rectangles could have been used to find the formulas for the squares. I can illustrate this thought by saying ‘the square is a special rectangle’. For example the formula for the T shape spacer required for a rectangular arrangement is as follows:
T = 2(W – 1) + 2(H – 1) now because the squares width and height are always the same, the formula could have been taken from this and it would read: T = 2(n - 1) + 2(n – 1) which would then read: T = 4n – 4, which is, if you refer back to page 3; the same formula used for the squares.
The same idea here can be said for the formula for the + shape spacers required for a rectangular tile arrangement. The formula is as follows; + = (W – 1) x (H – 1) the same idea once again is as follows: remembering that the height and width for a square is the same, the formula would then read: + = (n – 1)2 and by simply, once again referring back to page 3 will confirm that the formula used for the squares is too the same.
Finding the formulas for the rectangles by taking the idea from the squares was not so obvious. Now I feel that if I had have started this assignment with firstly investigating different rectangle designs, I now question myself; would I have then, have noticed what I have discovered? Was this why I was instructed to investigate different square arrangements first?
Tiling arrangements of different structures could be another investigation of interest. Take for example a T structure or even a cross structure. You could start of with very small diagrams and gradually make them bigger, display some data into a table and look for patterns.
How are your maths skills?
As the American author and philosopher Eric Hoffer says, “The hardest arithmetic to master is that which enables us to count our blessings.” That could be because we’re too busy counting what we don’t have, or what we feel we lack. But now there’s scientific evidence that counting your blessings is actually good for your health!
In one study, some college students were divided into two groups and instructed to keep a diary. One group was told to write about things they were thankful for each day, while the other group was told to write about any problems or simply routine events. Researchers found there were fewer illnesses among the thankful group, and that group also reported exercising more and being more supportive of others. Similar results were also found in other studies.
These experiments demonstrate the mind-body connection. Your thoughts can actually affect your body. When you are focused on the positive, counting the blessings in your life, your body naturally produces the “feel-good” chemical serotonin, resulting in a feeling of calm and well-being. But when you focus on your lack of the negative side of things, you are likewise draining your body of energy and no serotonin is released.
P.S I can see a pattern here, can you?
Claire McIlrath
